Page:Scientific Memoirs, Vol. 2 (1841).djvu/456

same, this peculiarity being solely referrible to the power of conduction $\chi$. If, for instance, $F$ designate, as was stated in § 4, the function, corresponding to such a case, of the dimensions and of the mean distance of both elements, the expression

$\alpha mm'(u'-u)\,d\,t$

not merely changes apparently into

$F(u'-u)\,d\,t$,

but also the equation

$\chi =\alpha mm's$

into the other,

$\chi =F\cdot s$,

(☉)

so that if we take the value of $F$ from this equation and place it in the above expression, we always obtain

${\frac {\chi (u'-u)\,d\,t}{s}}$.

Moreover, the circumstance of the expression (♂) still remaining valid for corpuscles, whose dimensions are no longer indefinitely small, is of some importance when the same electroscopic force only exists merely at all points of each such part. It is hence evident how intimately our considerations are allied to the spirit of the differential calculus; for uniformity in all points with reference to the property which enters into the calculation is precisely the distinctive characteristic required by the differential calculus from that which it is to receive as an element.

If we institute a more profound comparison between the process originating with Laplace and that here advanced, we shall arrive at some interesting points of comparison. If for instance we consider that for infinitely small masses at infinitely short distances all particular relations must necessarily have the same weight as for finite masses at finite distances, it is not directly evident how the method of the immortal Laplace—to whom we are indebted for so many valuable explanations respecting the nature of molecular actions,—according to which the elements must be constantly treated as if they were placed at finite distances from each other, could nevertheless still afford correct results; but we shall find on closer examination that it acts in fact otherwise than it expresses. Indeed, since Laplace, when determining the changes of an element by all surrounding it, makes the higher powers of the distance disappear compared with the lower, he therewith assumes, quite in the spirit of the